Theorem un0.1 45379

Description: ⊤ is the constant true, a tautology (see df-tru 1570). Kleene's "empty conjunction" is logically equivalent to ⊤. In a virtual deduction we shall interpret ⊤ to be the empty wff or the empty collection of virtual hypotheses. ⊤ in a virtual deduction translated into conventional notation we shall interpret to be Kleene's empty conjunction. If 𝜃 is true given the empty collection of virtual hypotheses and another collection of virtual hypotheses, then it is true given only the other collection of virtual hypotheses. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
un0.1.1	⊢ (   ⊤   ▶   𝜑   )
un0.1.2	⊢ (   𝜓   ▶   𝜒   )
un0.1.3	⊢ (   (   ⊤   ,   𝜓   )   ▶   𝜃   )

Assertion

Ref	Expression
un0.1	⊢ (   𝜓   ▶   𝜃   )

Proof of Theorem un0.1

Step	Hyp	Ref	Expression
1		un0.1.1	. . . 4 ⊢ (   ⊤   ▶   𝜑   )
2	1	in1 45172	. . 3 ⊢ (⊤ → 𝜑)
3		un0.1.2	. . . 4 ⊢ (   𝜓   ▶   𝜒   )
4	3	in1 45172	. . 3 ⊢ (𝜓 → 𝜒)
5		un0.1.3	. . . 4 ⊢ (   (   ⊤   ,   𝜓   )   ▶   𝜃   )
6	5	dfvd2ani 45184	. . 3 ⊢ ((⊤ ∧ 𝜓) → 𝜃)
7	2, 4, 6	uun0.1 45378	. 2 ⊢ (𝜓 → 𝜃)
8	7	dfvd1ir 45174	1 ⊢ (   𝜓   ▶   𝜃   )

Syntax hints:  ⊤wtru 1568  (   wvd1 45170  (   wvhc2 45181

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-vd1 45171  df-vhc2 45182

This theorem is referenced by:  sspwimpVD  45519